Sign Conventions

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH

Sign Conventions for Magnetic Fields

The MAD program uses the following Taylor expansion for the field on the mid-plane y=0, described in SLAC-75:

Note the factorial in the denominator. The field coefficients have the following meaning:

B₀: Dipole field, with a positive value in the positive y direction; a positive field bends a positively charged particle to the right.
B₁: Quadrupole coefficient
B₁ = (del B_y / del x);
a positive value corresponds to horizontal focussing of a positively charged particle.
B₂: Sextupole coefficient
B₂ = (del² B_y / del x²).
B₃: Octupole coefficient
B₃ = (del³ B_y / del x³).

Using this expansion and the curvature h of the reference orbit, the longitudinal component of the vector potential to order 4 is:

Taking curl A in curvilinear coordinates, the field components can be computed as

It can be easily verified that both curl B and div B are zero to the order of the B₃ term. Introducing the magnetic rigidity Brho, the multipole coefficients are computed as

K_n = e B_n / p_s = B_n / B rho.

hansg, June 17, 2002